Affineness is insensitive to thickenings

Proof. This is a special case of Limits, Proposition 31.11.2. $\square$

Proof for a finite order thickening. Suppose that $X \subset X'$ is a finite order thickening with $X$ affine. Then we may use Serre's criterion to prove $X'$ is affine. More precisely, we will use Cohomology of Schemes, Lemma 29.3.1. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_{X'}$-module. It suffices to show that $H^1(X', \mathcal{F}) = 0$. Denote $i : X \to X'$ the given closed immersion and denote $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp : \mathcal{O}_{X'} \to i_*\mathcal{O}_ X)$. By our discussion of finite order thickenings (following Definition 36.2.1) there exists an $n \geq 0$ and a filtration

$0 = \mathcal{F}_{n + 1} \subset \mathcal{F}_ n \subset \mathcal{F}_{n - 1} \subset \ldots \subset \mathcal{F}_0 = \mathcal{F}$

by quasi-coherent submodules such that $\mathcal{F}_ a/\mathcal{F}_{a + 1}$ is annihilated by $\mathcal{I}$. Namely, we can take $\mathcal{F}_ a = \mathcal{I}^ a\mathcal{F}$. Then $\mathcal{F}_ a/\mathcal{F}_{a + 1} = i_*\mathcal{G}_ a$ for some quasi-coherent $\mathcal{O}_ X$-module $\mathcal{G}_ a$, see Morphisms, Lemma 28.4.1. We obtain

$H^1(X', \mathcal{F}_ a/\mathcal{F}_{a + 1}) = H^1(X', i_*\mathcal{G}_ a) = H^1(X, \mathcal{G}_ a) = 0$

The second equality comes from Cohomology of Schemes, Lemma 29.2.4 and the last equality from Cohomology of Schemes, Lemma 29.2.2. Thus $\mathcal{F}$ has a finite filtration whose successive quotients have vanishing first cohomology and it follows by a simple induction argument that $H^1(X', \mathcal{F}) = 0$. $\square$

## Comments (3)

Comment #1089 by Nuno Cardoso on

When $i : X \to X'$ is a finite order thickening this result is an easy consequence of Serre's criterion for affineness. Since 31.11.2 does not seem to be an easy result, I wonder if it would be worthy to add the direct proof of this particular case here.

Comment #1094 by on

Thanks for this and your other comments. I have added a proof for the finite order case, see this commit.

There are many places where one can give easier arguments in special cases, or just easier arguments. Submissions of this kind are very welcome!

In the general set-up of the Stacks project it would make sense to have a separated lemma for the finite order case and move it to the section containing Serre's criterion with a forward link to this general result... This is a TODO.

Comment #3819 by slogan_bot on

Suggested slogan: Affineness is insensitive to thickenings

There are also:

• 4 comment(s) on Section 36.2: Thickenings

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06AD. Beware of the difference between the letter 'O' and the digit '0'.